Rational matrix factorizations via defect functors based on - - PowerPoint PPT Presentation
Rational matrix factorizations via defect functors based on 1005.2117 and 1112.XXXX Nicolas Behr Humboldt-Universitt zu Berlin/AEI in collaboration with Stefan Fredenhagen Max-Planck-Institute for Gravitational Physics (AEI) Maxwell
based on 1005.2117 and 1112.XXXX
Nicolas Behr
Humboldt-Universität zu Berlin/AEI
in collaboration with Stefan Fredenhagen
Max-Planck-Institute for Gravitational Physics (AEI)
Maxwell Institute, October 12th 2011
A little RCFT background Bulk correspondence Introducing (B-type) boundaries Data for boundary KS models Boundary LG theory data Preliminary version of RCFT/LG boundary correspondnce Defect functors
Witten, 1984
Gk ◮ class of CFTs that describe the motion of a string on a group manifold ◮ G Lie group, k ∈ Z>0 ”level” of the WZNW model ◮ action is of the form SWZNW = Skinetic + k · SWZ ◮ extraordinary features:
⊲ algebra of conserved currents = affine Lie algebra gk ⊲ primary fields labeled by highest weight representations of gk ⇒ finite number of primary fields, i.e. these theories are examples of rational CFTs
◮ Construction:
Kazama and Suzuki, 1989
supersymmetrize
− − − − − − − − − → N = 1 version
gauge subgroup
− − − − − − − − − → WZNW coset
Gk H × SO(2d)1
with:
⊲ G simple compact Lie group ⊲ k level of the corresponding affine Lie algebra gk ⊲ H ⊂ G regularly embedded subgroup (i.e. rk G = rk H) ⊲ 2d = dim G − dim H
Note: the Majorana-fermions are realized in "bosonized form", i.e. as a so(2d)1 WZNW-model ◮ Motivation: provides a large class of N = (2, 2) rational SCFTs
SU(n + k)1 × SO(2nk)1 SU(n)k+1 × SU(k)n+1 × U(1) ∼ = SU(n + 1)k × SO(2n)1 SU(n)k+1 × U(1) ◮ Note: we use the diagram embedding
. . . SU(n) SU(n + 1)
SU(n + k)1 × SO(2nk)1 SU(n)k+1 × SU(k)n+1 × U(1) ∼ = SU(n + 1)k × SO(2n)1 SU(n)k+1 × U(1) ◮ Note: we use the diagram embedding
. . . SU(n) SU(n + 1)
i(h, ζ) = hζ ζ−n
h ∈ SU(n), ζ ∈ U(1) Since i(ξ−11, ξ) = 1 for ξn = 1, "H ⊂ Gk " only if we quotient by the Zn action: U(n) =
⇒ field identifications!
SU(n + 1)k/U(n) ≡ SU(n + 1)k × SO(2n)1 SU(n)k+1 × U(1) ◮ highest weight labels: ( Λ
, Σ
; λ
, µ
) where the so(2d)1 for any d can take values
⊲ Σ = 0, v : Neveu-Schwarz sector ⊲ Σ = s, s Ramond sector
SU(n + 1)k/U(n) ≡ SU(n + 1)k × SO(2n)1 SU(n)k+1 × U(1) ◮ highest weight labels: ( Λ
, Σ
; λ
, µ
) where the so(2d)1 for any d can take values
⊲ Σ = 0, v : Neveu-Schwarz sector ⊲ Σ = s, s Ramond sector
◮ non-trivial common center Z = i−1(ZSU(n+1)) of the numerator and denominator theory ⇒ cyclic group Zn(n+1) (simple currents) Gid
SU(n + 1)k/U(n) ≡ SU(n + 1)k × SO(2n)1 SU(n)k+1 × U(1) ◮ highest weight labels: ( Λ
, Σ
; λ
, µ
) where the so(2d)1 for any d can take values
⊲ Σ = 0, v : Neveu-Schwarz sector ⊲ Σ = s, s Ramond sector
◮ non-trivial common center Z = i−1(ZSU(n+1)) of the numerator and denominator theory ⇒ cyclic group Zn(n+1) (simple currents) Gid ◮ labels are restricted by
Gepner, 1989; Lerche et al., 1989; Moore and Seiberg, 1989
⊲ identification rules via action of Gid,
Schellekens and Yankielowicz, 1989, 1990
generated by the simple current J0 = (Jn+1, v; Jn, k + n) (Λ, Σ; λ, µ) ∼ Jm
0 (Λ, Σ; λ, µ)
∀m ∈ Z ⊲ selection rules: monodromy charges of the numerator and denominator parts should be equal QJn+1(Λ) + Qv(Σ)
!
= QJn(λ) + Qk+n(µ) with QJ(φ) = hJ + hφ − hJφ mod 1
A little RCFT background Bulk correspondence Introducing (B-type) boundaries Data for boundary KS models Boundary LG theory data Preliminary version of RCFT/LG boundary correspondnce Defect functors
choice of W
Idea: {ring of chiral prim. fields} ↔ fusion ring ◮ chiral primary fields: h = q
2 and h = q 2
◮ OPE of chiral primary fields: Φ(z)Υ(z′) ∼ . . . + 1 (z − z′)hΦ+hΥ−hΦΥ (ΦΥ)(z) + . . .
choice of W
Idea: {ring of chiral prim. fields} ↔ fusion ring ◮ chiral primary fields: h = q
2 and h = q 2
◮ OPE of chiral primary fields: Φ(z)Υ(z′) ∼ . . . + 1 (z − z′)hΦ+hΥ−hΦΥ (ΦΥ)(z) + . . . ◮ since hΦΥ ≥ (qΦ + qΥ)/2 = hΦ + hΥ, we obtain, rescaling coordinates by λ and taking the limit λ → ∞: Φ(z)Υ(z) := lim
z′→z Φ(z)Υ(z′) =
0 else ⇒ ring of chiral primary fields
choice of W
Idea: {ring of chiral prim. fields} ↔ fusion ring ◮ chiral primary fields: h = q
2 and h = q 2
◮ OPE of chiral primary fields: Φ(z)Υ(z′) ∼ . . . + 1 (z − z′)hΦ+hΥ−hΦΥ (ΦΥ)(z) + . . . ◮ since hΦΥ ≥ (qΦ + qΥ)/2 = hΦ + hΥ, we obtain, rescaling coordinates by λ and taking the limit λ → ∞: Φ(z)Υ(z) := lim
z′→z Φ(z)Υ(z′) =
0 else ⇒ ring of chiral primary fields ◮ Gepner: cpf ring is the same as a truncation of the fusion ring C Λ1 × C Λ2 = f (su(n+1)) Λ
Λ1 Λ2
f (su(n)) PΛ
PΛ1 PΛ2
δ(Q − Q1 − Q2)C Λ
choice of W
Idea: {ring of chiral prim. fields} ↔ fusion ring ◮ chiral primary fields: h = q
2 and h = q 2
◮ OPE of chiral primary fields: Φ(z)Υ(z′) ∼ . . . + 1 (z − z′)hΦ+hΥ−hΦΥ (ΦΥ)(z) + . . . ◮ since hΦΥ ≥ (qΦ + qΥ)/2 = hΦ + hΥ, we obtain, rescaling coordinates by λ and taking the limit λ → ∞: Φ(z)Υ(z) := lim
z′→z Φ(z)Υ(z′) =
0 else ⇒ ring of chiral primary fields ◮ Gepner: cpf ring is the same as a truncation of the fusion ring C Λ1 × C Λ2 = f (su(n+1)) Λ
Λ1 Λ2
f (su(n)) PΛ
PΛ1 PΛ2
δ(Q − Q1 − Q2)C Λ ◮ Our paper: explicit computation of the SU(3)k/U(2) fusion ring via relation generating potential ⇒ Wk(y1, y2)
bulk LG-Action: a theory of chiral scalar superfields
SLG =
⊲ K(Φ, Φ) Kähler potential ⊲ W (Φ) superpotential ⊲ theory flows to CFT in IR ⇔ W (Φ) is quasihomogeneous: W (eiλqi Φi) = e2iλW (Φi) ∀λ ∈ C
bulk LG-Action: a theory of chiral scalar superfields
SLG =
⊲ K(Φ, Φ) Kähler potential ⊲ W (Φ) superpotential ⊲ theory flows to CFT in IR ⇔ W (Φ) is quasihomogeneous: W (eiλqi Φi) = e2iλW (Φi) ∀λ ∈ C
◮ Question: How do we choose W (Φi)? Answer: for our purposes (Grassmannian Kazama-Suzuki models), employ Gepner’s method, i.e. use the polynomial W (Φi) such that chiral ring of KS model = JacW (Φi ) := C[Φi] ∂iW , which implies that a given chiral primary state Λcp is associated to some explicit polynomial UΛ(Φi) ∈ JacW (Φi ).
A little RCFT background Bulk correspondence Introducing (B-type) boundaries Data for boundary KS models Boundary LG theory data Preliminary version of RCFT/LG boundary correspondnce Defect functors
◮ bulk Hilbert space: "almost diagonal" modular invariant H =
H[Λ,Σ;λ,µ] ⊗ H[Λ,Σ+;λ,µ]
◮ bulk Hilbert space: "almost diagonal" modular invariant H =
H[Λ,Σ;λ,µ] ⊗ H[Λ,Σ+;λ,µ] ◮ boundary Hilbert space: via folding trick ⇒ theory on upper half plane w/ bdry at the real line z = z, where we demand B-type gluing conditions: T(z) = T(z) J(z) = J(z) G ±(z) = ηG
±(z)
Imz = Imz with: η a sign corresponding to the choice of a spin structure, i.e. of GSO projection
◮ bulk Hilbert space: "almost diagonal" modular invariant H =
H[Λ,Σ;λ,µ] ⊗ H[Λ,Σ+;λ,µ] ◮ boundary Hilbert space: via folding trick ⇒ theory on upper half plane w/ bdry at the real line z = z, where we demand B-type gluing conditions: T(z) = T(z) J(z) = J(z) G ±(z) = ηG
±(z)
Imz = Imz with: η a sign corresponding to the choice of a spin structure, i.e. of GSO projection ◮ B-type D-branes via Cardy construction and factorisation into twisted boundary sectors
Fredenhagen, 2003; Ishikawa, 2002; Ishikawa and Tani, 2003, 2004
|L, S; l = N
ψ(n+1)
LΛ
S(so)
SΣ ψ (n) lλ
0Λ
S(so)
0Σ S(n) 0λ
|Λ, Σ; λ, 0
article by Stanciu [1998] ◮ Cardy branes are the maximally symmetric types of D-brane solutions, i.e. satisfy the much more restrictive gluing conditions Wi(z) = ω(W i)(z),
Cardy, 1989
with
⊲ Wi(z) chiral algebra current ⊲ ω outer automorphism of the chiral algebra
⇒ Cardy branes preserve not just the N = 2 symmetry, but the full chiral algebra A on the boundary!
◮ introducing boundary breaks translation invariance normal to bdry ⇒ at least half of the N = (2, 2) symmetry broken
◮ introducing boundary breaks translation invariance normal to bdry ⇒ at least half of the N = (2, 2) symmetry broken ◮ case W = 0: SUSY-variation of SLG yields surface term that can be compensated by adding Sbdry (in bulk fields) to SLG
◮ introducing boundary breaks translation invariance normal to bdry ⇒ at least half of the N = (2, 2) symmetry broken ◮ case W = 0: SUSY-variation of SLG yields surface term that can be compensated by adding Sbdry (in bulk fields) to SLG ◮ W = 0: SUSY-variation of SLG + Sbdry results in term δ (SLG + Sbdry) = i 2
′
π (∗) that can not be compensated by contributions to SLG in bulk fields (Warner problem)
Warner, 1995
◮ introducing boundary breaks translation invariance normal to bdry ⇒ at least half of the N = (2, 2) symmetry broken ◮ case W = 0: SUSY-variation of SLG yields surface term that can be compensated by adding Sbdry (in bulk fields) to SLG ◮ W = 0: SUSY-variation of SLG + Sbdry results in term δ (SLG + Sbdry) = i 2
′
π (∗) that can not be compensated by contributions to SLG in bulk fields (Warner problem)
Warner, 1995
◮ way out: introduce boundary fermionic superfield Π ≡ Π(s, θ0, θ
0) = π(s) + . . . + θ 0(E(Φ) + . . .) with ”LG-like” action
SΠ = −1 2
0 − i
2
0 + c.c.
⇒ SUSY-variation of SΠ cancels (∗) iff
Brunner et al., 2003; Kapustin and Li, 2003; Kontsevich; Orlov, 2003
W = J · E + const = matrix factorization!
◮ Cardy branes = maximally symmetric D-branes, i.e. preerve N = 2, but also the full chiral symmetry A
◮ Cardy branes = maximally symmetric D-branes, i.e. preerve N = 2, but also the full chiral symmetry A ◮ ”LG theory D-branes”, i.e. matrix factorizations, were constructed to preserve N = 2 explicitly
◮ Cardy branes = maximally symmetric D-branes, i.e. preerve N = 2, but also the full chiral symmetry A ◮ ”LG theory D-branes”, i.e. matrix factorizations, were constructed to preserve N = 2 explicitly
A little RCFT background Bulk correspondence Introducing (B-type) boundaries Data for boundary KS models Boundary LG theory data Preliminary version of RCFT/LG boundary correspondnce Defect functors
◮ bulk Hilbert space: H =
[Λ,Σ;λ,µ] H[Λ,Σ;λ,µ] ⊗ H[Λ,Σ+;λ,µ]
◮ (B-type) Cardy branes: | L
k+3
, S
; ℓ
= N
ψ(3)
LΛ S(so) SΣ S (2) ℓλ
0Λ S(so) 0Σ S(2) 0λ
|Λ, Σ; λ, 0
affine Lie-algebra, while the symbol S stands for the regular modular S-matrices ◮ |L, v; ℓ = |L, 0; ℓ ⇒ shorthand notation: |L, ℓ ≡ |L, 0; ℓ ◮ spectra of (chiral primary) open strings can be computed from L1, l1| q
1 2 (L0+L0− c 12 )|L2, l2ch.prim.
=
nΛL2
L1N(k+1) Λ1l2 l1χΛ,0;Λ1,Λ1+2Λ2(q)
◮ B-type D-branes couple only to uncharged RR ground states! ◮ SU(3)k/U(2) models: cpf = {(Λ1, Λ2), 0; Λ1, Λ1 + 2Λ2)}
spectral flow
− − − − − − → RGS
uncharged
− − − − − → RGS0 = [j] = {[(j, j), s; 2j + 1, 0]}
◮ B-type D-branes couple only to uncharged RR ground states! ◮ SU(3)k/U(2) models: cpf = {(Λ1, Λ2), 0; Λ1, Λ1 + 2Λ2)}
spectral flow
− − − − − − → RGS
uncharged
− − − − − → RGS0 = [j] = {[(j, j), s; 2j + 1, 0]} ⇒ RR-charge chj(|L, ℓ) is given by coefficient of [j] in the formula |L, S; ℓ = N
ψ(3)
LΛ S(so) SΣ S (2) ℓλ
0Λ S(so) 0Σ S(2) 0λ
|Λ, Σ; λ, 0
◮ B-type D-branes couple only to uncharged RR ground states! ◮ SU(3)k/U(2) models: cpf = {(Λ1, Λ2), 0; Λ1, Λ1 + 2Λ2)}
spectral flow
− − − − − − → RGS
uncharged
− − − − − → RGS0 = [j] = {[(j, j), s; 2j + 1, 0]} ⇒ RR-charge chj(|L, ℓ) is given by coefficient of [j] in the formula |L, S; ℓ = N
ψ(3)
LΛ S(so) SΣ S (2) ℓλ
0Λ S(so) 0Σ S(2) 0λ
|Λ, Σ; λ, 0
chj(|L, l) = N ψ(3)
L (j,j)Sso 0s S(2) l 2j+1
(0,0)(j,j)Sso 0s S(2) 0 2j+1
◮ B-type D-branes couple only to uncharged RR ground states! ◮ SU(3)k/U(2) models: cpf = {(Λ1, Λ2), 0; Λ1, Λ1 + 2Λ2)}
spectral flow
− − − − − − → RGS
uncharged
− − − − − → RGS0 = [j] = {[(j, j), s; 2j + 1, 0]} ⇒ RR-charge chj(|L, ℓ) is given by coefficient of [j] in the formula |L, S; ℓ = N
ψ(3)
LΛ S(so) SΣ S (2) ℓλ
0Λ S(so) 0Σ S(2) 0λ
|Λ, Σ; λ, 0
chj(|L, l) = N ψ(3)
L (j,j)Sso 0s S(2) l 2j+1
(0,0)(j,j)Sso 0s S(2) 0 2j+1
◮ basis: in terms of charges of the |L, 0 branes chj(|L, l) =
⌊ k
2 ⌋
LL′ l − N(k+1) LL′ k+1−l
chj(|L′, 0)
CFT flow rules
Fredenhagen, 2003; Fredenhagen and Schomerus, 2003
flow induced by tachyon Ψ∗ = (Ψa
∗, Ψb ∗)
|L, ℓ − 1 + |L, ℓ Ψa
∗
+ |L, ℓ + 1 Ψb
∗
K=L−1|K, ℓ
for L = k
2
|L − 1, ℓ for L = k
2
Important: Ψ∗ has a specific U(1)-R-charge qΨ∗ (= 1/(k + 3))!
Topological defects
here: consider as operators DΘ ≡ D[(Λ1,Λ2),Σ;λ,µ] that ◮ form a semi-ring under ”fusion” ∗ DΘ1 ∗ DΘ2 =
n
Θ Θ1Θ2 DΘ
(n
Θ Θ1Θ2
∈ Z≥0) ◮ act on Cardy branes B|L,ℓ (resulting in new Cardy branes) ◮ Most important feature: ∃ defect DΘ(1) that generates all Cardy branes from the |L, 0 branes via DΘ(1) ∗ B|L,ℓ = B|L,ℓ−1 + B|L,ℓ+1
A little RCFT background Bulk correspondence Introducing (B-type) boundaries Data for boundary KS models Boundary LG theory data Preliminary version of RCFT/LG boundary correspondnce Defect functors
Let R be a graded polynomial ring, i.e. R ≡ C[yi]gr := ⊕i∈Z≧0Ri ; ∀p ∈ Ri : deg(p) = i with deg(yi) = wi ∈ N. Let Wk(yi) ∈ Rk+3 a quasihomogeneous
Wk(eiλqi yi)
!
= e2iλWk(yi) ∀λ ∈ C∗ induces a U(1)-R-charge grading qyi = 2wi/(k + 3).
Definition: category hmf gr(Wk) ◮ Ob(hmf gr(Wk)) :=
⊲ Q = J E
⊲ Q2 = J · E J · E
⊲ σ · Q · σ = −Q (σ2 = −12r×2r) ⊲ ρ(λ; yi)Q(eiλqi yi)ρ−1(λ; yi) = eiλQ(yi) ∀λ ∈ C∗
Definition: category hmf gr(Wk) ◮ Ob(hmf gr(Wk)) :=
⊲ Q = J E
⊲ Q2 = J · E J · E
⊲ σ · Q · σ = −Q (σ2 = −12r×2r) ⊲ ρ(λ; yi)Q(eiλqi yi)ρ−1(λ; yi) = eiλQ(yi) ∀λ ∈ C∗
◮ Mor(hmf gr(Wk)) :=
σBΦσA = (−1)|Φ|Φ (| Φ |∈ Z2) QBΦ − (−1)|Φ|ΦQA = 0 mod
Ψ + (−1)|Φ| ΦQA ρB(λ; yi)Φ(eiλqi yi)ρ−1
A (λ; yi) = eiλqΦ(yi)
i.e. this is the definition of some (graded) cohomology of MFs ◮ composition of morphisms: composition in cohomological sense (i.e. naive composition up to exactness)
Definition Q ∼ Q′ :⇔ w.l.o.g rk(Q) ≤ rk(Q′) Q′ = U
triv Qt⊕n triv
where ◮ m, n ∈ Z≥0 s.th. rk(Q) + m + n = rk(Q′) ◮ Qtriv =
Wk
U1 U2
(r ′ = rk(Q′)) i.e. U is invertible over R: UU−1 = U−1U = 12r ′×2r ′ severe technical difficulty: equivalences make it hard to guess ”interesting” MFs!
via Kapustin-Li formula:
Kapustin and Li, 2004
chφ(Q) = 1 √ 2 ResWk
where φ ∈ JacW (i.e. some polynomial in y1 and y2), Q is a MF and Str denotes the supertrace, while the residue is formally defined as ResWk(f ) = 1 (2πi)2 f ∂y1Wk∂y2Wk dy1dy2
via Kapustin-Li formula:
Kapustin and Li, 2004
chφ(Q) = 1 √ 2 ResWk
where φ ∈ JacW (i.e. some polynomial in y1 and y2), Q is a MF and Str denotes the supertrace, while the residue is formally defined as ResWk(f ) = 1 (2πi)2 f ∂y1Wk∂y2Wk dy1dy2 ◮ Note: to compare this with the CFT RR charges, we need the explicit ”dictionary” between the elements of the CFT and of the LG theory chiral rings: Λcpf ≡ [(Λ1, Λ2), 0; Λ1, Λ1 + 2Λ2]
⌊Λ1/2⌋
(−1)r Λ1 − r r
1
y Λ2+r
2
Def.: operator τ : Hi,q( RQA, RQB) :
Φ
− → QB
τΦ
− − → QB
Def.: operator τ : Hi,q( RQA, RQB) :
Φ
− → QB
τΦ
− − → QB
◮ Def.: shift functor [1]
⊲ Q = J E
−E −J
φ0 φ1
→ Φ[1] := φ1 φ0
Def.: operator τ : Hi,q( RQA, RQB) :
Φ
− → QB
τΦ
− − → QB
◮ Def.: shift functor [1]
⊲ Q = J E
−E −J
φ0 φ1
→ Φ[1] := φ1 φ0
◮ Def.: cone functor c
⊲ c
Φ
− → QB
JΦ EΦ
JB τφ0 −EA EB τφ1 −JA ⊲ on diagrams commutative up to exact morphisms A:
c QA
f
g′
f ′
QD := c(f)
c(g,h;a)
c(g, h; a)i := hi ai 0 gi+1
◮ generate new MFs via c(QA
Φ
− → QB)
◮ generate new MFs via c(QA
Φ
− → QB) ◮ Def.: distinguished triangles
⊲ (TR1) ∀Φ ∈ H0,q( RQA, RQB)∃ distinguished △ QA
Φ
− → QB
p(Φ)
− − → c(Φ)
q(Φ)
− − → QA[1] p(Φ) := 1B
QB
p(Φ) c(Φ) q(Φ) p(p(Φ))
Φ[1] QB[1]
c(p(Φ))
∼ = ∃
◮ generate new MFs via c(QA
Φ
− → QB) ◮ Def.: distinguished triangles
⊲ (TR1) ∀Φ ∈ H0,q( RQA, RQB)∃ distinguished △ QA
Φ
− → QB
p(Φ)
− − → c(Φ)
q(Φ)
− − → QA[1] p(Φ) := 1B
QB
p(Φ) c(Φ) q(Φ) p(p(Φ))
Φ[1] QB[1]
c(p(Φ))
∼ = ∃
c QA[−1]
τ
QC → QA
c(g,0;a)
⇒ may generate complicated cones from simpler MFs!
A little RCFT background Bulk correspondence Introducing (B-type) boundaries Data for boundary KS models Boundary LG theory data Preliminary version of RCFT/LG boundary correspondnce Defect functors
Wk(x) = xk+2 ◮ easiest matrix factorizations: polynomial MFs: Qi =
xk+2−i
Wk(x) = xk+2 ◮ easiest matrix factorizations: polynomial MFs: Qi =
xk+2−i
◮ analysis of spectra and U(1)-R- and RR-charges in both theories leads to the association
Brunner et al., 2003; Kapustin and Li, 2003
⇒ Complete solution!
Wk(y1, y2) =
⌊ k+1
2 ⌋
(y 2
1 − βjy2) ·
y1 for k even 1 for k odd where βj = 2
d
|L, 0 ↔ Q|L,0 =
j=0(y 2 1 − βjy2) Wk L
j=0(y 2 1 −βj y2)
Wk(y1, y2) =
⌊ k+1
2 ⌋
(y 2
1 − βjy2) ·
y1 for k even 1 for k odd where βj = 2
d
|L, 0 ↔ Q|L,0 =
j=0(y 2 1 − βjy2) Wk L
j=0(y 2 1 −βj y2)
unlike all branes |L, ℓ with ℓ > 0. But all polynomial MFs have no fermions in their self-spectra ⇒ need to construct higher-rank MFs!
Wk(y1, y2) =
⌊ k+1
2 ⌋
(y 2
1 − βjy2) ·
y1 for k even 1 for k odd where βj = 2
d
|L, 0 ↔ Q|L,0 =
j=0(y 2 1 − βjy2) Wk L
j=0(y 2 1 −βj y2)
unlike all branes |L, ℓ with ℓ > 0. But all polynomial MFs have no fermions in their self-spectra ⇒ need to construct higher-rank MFs! ◮ available data:
⊲ number of bosonic/fermionic open strings in all spectra ⊲ U(1)-R-charges of these open-strings in the spectra ⊲ RR-chages carried by the D-branes resp. matrix factorizations
Wk(y1, y2) =
⌊ k+1
2 ⌋
(y 2
1 − βjy2) ·
y1 for k even 1 for k odd where βj = 2
d
|L, 0 ↔ Q|L,0 =
j=0(y 2 1 − βjy2) Wk L
j=0(y 2 1 −βj y2)
unlike all branes |L, ℓ with ℓ > 0. But all polynomial MFs have no fermions in their self-spectra ⇒ need to construct higher-rank MFs! ◮ available data:
⊲ number of bosonic/fermionic open strings in all spectra ⊲ U(1)-R-charges of these open-strings in the spectra ⊲ RR-chages carried by the D-branes resp. matrix factorizations ⊲ specifically for the SU(3)k/U(2) model: CFT flow rules
BCFT flow rule |L, 0 + |L, 1 Ψ∗ ⊕L+1
K=L−1|K, 0
BCFT flow rule |L, 0 + |L, 1 Ψ∗ ⊕L+1
K=L−1|K, 0
"Translating" this into a LG-theory triangle, we obtain: . . . → Q|L,0[1]
ψ∗
− − → Q|L,1 → ⊕L+1
K=L−1Q|K,0 → Q|L,0[2] → . . .
where we know the MFs colored in green and that the triangle is distinguished for any given morphism ψ∗, whence this allows us to shift the triangle to obtain a candidate for Q|L,1: Q|L,1
?
= c
K=L−1Q|K,0[−1] → Q|L,0[1]
BCFT flow rule |L, 0 + |L, 1 Ψ∗ ⊕L+1
K=L−1|K, 0
"Translating" this into a LG-theory triangle, we obtain: . . . → Q|L,0[1]
ψ∗
− − → Q|L,1 → ⊕L+1
K=L−1Q|K,0 → Q|L,0[2] → . . .
where we know the MFs colored in green and that the triangle is distinguished for any given morphism ψ∗, whence this allows us to shift the triangle to obtain a candidate for Q|L,1: Q|L,1
?
= c
K=L−1Q|K,0[−1] → Q|L,0[1]
ψ∗ of the correct U(1)-R-charge, which leads to: |L, 1 = Q|L,1 = c(⊕L+1
K=L−1Q|K,0)[−1] φ∗
Via SINGULAR code for the explicit computation of H1(QA, QB) for any MFs Qi (thanks to N. Carqueville for initial code!), we can pursue the brute force Ansatz polynomial MFs H1,q(QA, QB) cones of polynomial MFs cones of cones
MFs . . . My code allows to compute the explicit spectra for all such MFs, i.e. we can search for suitable MFs ⇒ confirmation of the previously shown MFs, some sporadic mathces for higher label branes |L, ℓ with ℓ > 1
A little RCFT background Bulk correspondence Introducing (B-type) boundaries Data for boundary KS models Boundary LG theory data Preliminary version of RCFT/LG boundary correspondnce Defect functors
The superpotential of the SUk(3)/U(2) KS model can be expressed as Wk(y1, y2) =
1
+ xk+3
2
x1x2→y2
The superpotential of the SUk(3)/U(2) KS model can be expressed as Wk(y1, y2) =
1
+ xk+3
2
x1x2→y2
This may be seen from a graphical representation:
αd (0) (1) (2) (Lmax − 1) (Lmax) (Lmax − 1)−1 (2)−1 (1)−1 (0)−1
W(xi) =
j (x1 − eiαLx2)
group symmetric factors
(0) (1) (2) (Lmax − 1) (Lmax)
x1 + x2 = y1
W(yi) = y1
1 − (2 + eiαL + e−iαL)y2)
The superpotential of the SUk(3)/U(2) KS model can be expressed as Wk(y1, y2) =
1
+ xk+3
2
x1x2→y2
This may be seen from a graphical representation:
αd (0) (1) (2) (Lmax − 1) (Lmax) (Lmax − 1)−1 (2)−1 (1)−1 (0)−1
W(xi) =
j (x1 − eiαLx2)
group symmetric factors
(0) (1) (2) (Lmax − 1) (Lmax)
x1 + x2 = y1
W(yi) = y1
1 − (2 + eiαL + e−iαL)y2)
Also note the Zk+3 rotation symmetry!
Definition: Let R and S be two (graded) polynomial rings, R − mod and S − mod the categories of left R- resp. S-modules and homomorphisms, and let Φ : R → S be a ring homomorphism. Then the pullback and pushforward functors along Φ Φ∗ : R − mod ⇆ S − mod : Φ∗ as follows: Φ∗ :
→ S ⊗R X ∈ SM f ∈ Mor(R − mod) → 1S ⊗
R f ∈ Mor(S − mod)
Φ∗ : via ∀r ∈ R, x ∈ X, X ∈ SM : r.x := Φ(r).x and analogously for morphisms
Definition: Let R and S be two (graded) polynomial rings, R − mod and S − mod the categories of left R- resp. S-modules and homomorphisms, and let Φ : R → S be a ring homomorphism. Then the pullback and pushforward functors along Φ Φ∗ : R − mod ⇆ S − mod : Φ∗ as follows: Φ∗ :
→ S ⊗R X ∈ SM f ∈ Mor(R − mod) → 1S ⊗
R f ∈ Mor(S − mod)
Φ∗ : via ∀r ∈ R, x ∈ X, X ∈ SM : r.x := Φ(r).x and analogously for morphisms ◮ Note: for suitable choices of Φ, these functors naturally act on MFs and morphisms of MFs!
◮ Ansatz: Consider the ring homomprhisms realizing Wk(yi) → Wk(xi) = xk+3
1
+ xk+3
2
and the morphism that generates the Zk+3 rotation: ι : R ≡ C[yi] → S ≡ C[xi] :
y2 → x1x2 γk : S → S :
x2 → e2iπ/(k+3)x2
◮ Ansatz: Consider the ring homomprhisms realizing Wk(yi) → Wk(xi) = xk+3
1
+ xk+3
2
and the morphism that generates the Zk+3 rotation: ι : R ≡ C[yi] → S ≡ C[xi] :
y2 → x1x2 γk : S → S :
x2 → e2iπ/(k+3)x2 ◮ The functor D(1) defined as
Behr and Fredenhagen, 2011
R − mod
D(1)
S − mod
(γk)∗ S − mod ι∗
◮ Ansatz: Consider the ring homomprhisms realizing Wk(yi) → Wk(xi) = xk+3
1
+ xk+3
2
and the morphism that generates the Zk+3 rotation: ι : R ≡ C[yi] → S ≡ C[xi] :
y2 → x1x2 γk : S → S :
x2 → e2iπ/(k+3)x2 ◮ The functor D(1) defined as
Behr and Fredenhagen, 2011
R − mod
D(1)
S − mod
(γk)∗ S − mod ι∗
functors”, according to D(1) ◦ D(n) = D(n−1) ⊕ D(n+1)
With the help of the ”defect functors” D(n), we can generate all ”rational” MFs from the simplest MFs Q|L,0:
With the help of the ”defect functors” D(n), we can generate all ”rational” MFs from the simplest MFs Q|L,0:
Checks: ◮ RR charges are automatically correct, since we may act with D(n) on the triangle describing Q|L,1, thereby obtaining c
K=L−1Q|K,ℓ[−1] D(ℓ)Φ(1)
− − − − − → Q|L,ℓ[1]
= Q|L,ℓ−1 ⊕ Q|L,ℓ+1 , which by induction yields the correct result in comparison with the RCFT data ◮ ∃ method (NBSF) to generate the correct U(1)-R-charge representations ρ|L,ℓ via D(ℓ) directly from the (unambiguously defined) rep ρ|L,0 ◮ explicit computations via SINGULAR for a large number of examples show agreement of spectra including the U(1)-R-charges
◮ apply method to other KS models, e.g. the SUk(N + 1)/U(N) Grassmannian models with Wk,N(y1, . . . , yN−1) := N−1
xk+N+1
i
◮ ∃ deformations of the SUk(3)/U(2) model that leave the defect functor semi-ring invariant or at least partially preserve it? ◮ relation to conventional ”defect technology” for LG theories: obtain defefct MFs via (Φ : R → S)
RDS := (Φ∗, 1) S1S S
DR := (Φ∗, 1) R1R ⇒ new insights in the classes of physically relevant topological defects for LG theories! ◮ potential application: Khovanov-Rozanski link homology computations
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